Analytical approach of the symmetry: Sharp supercritical Hardy–Sobolev inequalities and applications. (June 2018)

• Record Type: Journal Article
• Authors: Cotsiolis, Athanase; Labropoulos, Nikos
• Is Part Of: Nonlinear analysis. Volume 171(2018)
• Page Start: 134
• Page End: 155
• Publication Date: 2018-06
• Subjects: 41A44 -- 35B33 -- 58D19 -- 34B18; Analytical approach of the symmetry -- Supercritical Hardy–Sobolev inequalities -- Supercritical Hardy–Sobolev elliptic equations; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2018.02.005
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider the optimal Hardy–Sobolevinequality on smooth bounded symmetric domains of the Euclidean space without any assumption concerning the "shape" of the boundary (i.e. some convexity) confirming that the symmetry of a domain is an intrinsic property characterizing both the domain itself and its boundary. Our model domain is the solid torus because of its particular interest in terms both of the geometry and of the analysis. We apply the results of the above study to solve the supercritical problem (P) Δ p u + a ( x ) u p − 1 = f ( x ) u p ∗ ( s ) − 1 | x | s, u > 0 on T, u = 0 on ∂ T, 1 < p < 2, 0 ≤ s ≤ p and p ∗ ( s ) = p ( 2 − s ) 2 − p and some variants of it.